TL;DR: There's a lot more to the craft than just wires and a board. This started as a short post, but turned into Light Shows 101.
I've been on the fringes of the community for a few years. If you're willing to touch a soldering gun, you can do far better than a single-board computer, though I'm sure I'm going to annoy some Slashdotters for suggesting such blasphemy.
First, consider your requirements: Lights, elements, power, control, sequencing, and sound. We'll tackle them in that order.
Lights are actually one of the tougher decisions to get started with. You have to decide between LED (brilliant intensity, saturated colors, low power, expensive) or incandescent (cheap, high power, warm glow), and what size you'll use (everything from surface-mount LEDs to full-size Edison bulbs). I know a guy who's used car headlights in a show, and there's been a few stage lights, too.
The problem with lights is the sheer number to be used. 32 strings are enough for one megatree (more on megatrees shortly). Consider that shows are measured in the thousands of individual lights, and the hundreds of channels. Mixing LED and incandescent lights is rarely a good idea, because of the severe difference in brightness. Even among the same type, different brands have different colored lights. Manufacturers also tend to cut the strings shorter every year, saving money on wire costs. What used to be a 30-foot string several years ago is now a 27-foot string. The result is that show builders will hoard hundreds of strings of lights from the same manufacturer and year, so any future element projects will have a consistent look. Communities will also organize occasional bulk purchases directly from manufacturers, ordering a whole shipping container full of lights.
The elements, though, are where creativity and construction skill really comes into play. You're effectively building a piece of modern art that usually has to survive snow, rain, wind, squirrels (who apparently love the taste of wire insulation), vandals (sadly), and both heat and cold. The elements start the show cold, but often (especially with incandescent lights) they'll have so much current running through them that by the end of the night they're warm enough to cause noticeable expansion.
There are a few standard elements, but every builder has their own technique. There are basic trees, where lights make a vaguely conical shape. With more wiring, you get megatrees, which have several steps of control, up to a single string per control channel, so you can animate spinning motions. In a smaller size, tomato cages with their legs welded together make great minitrees. A long piece of conduit, wrapped in lights (about 30 feet of lights per foot of conduit) can be bent and anchored to stakes, making arches. Chicken wire and zip ties are the perfect tools for laying out a more two-dimensional element. With a bigger budget, flamethrowers, projectors, lasers, water fountains, and robotic spotlights are all options.
With all those elements, electric power becomes a problem. LEDs make the problem much more manageable, but there's still the issue of distribution. There's a lot of extension cords involved, and possibly some very thick wires needed. Builders of big shows will often turn to buried cables and upgraded supply lines. I helped with a show that used a 100-amp line running to a buried box in the back yard, where a control box fed a set of distributor boxes through 50-foot 20-amp lines that were scattered around the area supplying the other elements.
Every outdoor circuit must be also protected by a ground fault circuit interrupter (GFCI), which will cut the power immediately if it shorts to ground. It's intended to protect a person who accidentally grounds a circuit, but with that many connectors out in the rain and snow, you can expect a GFCI to trip a few times a week.
Controlling that much power is the part that gets so much attention, but it's really one of the easiest aspects of the hobby. There are a few routes to go, from the aforementioned newbie-friendly Light-O-Rama to all-custom rigs like you describe. The best option depends on the elements and lights in use. LOR has its own proprietary system, but it can also run on the DMX protocol, which is well-established in the theatre industry. For addressable RGB LEDs, or big fields of pixels, modern stage control systems (not DMX) are better suited for the huge numbers of individual controls needed.
Personally, I like the Cat5-cable-based Lynx series, built by the fine folks at DiyLightAnimation. The community runs co-op purchases throughout the year, getting bulk rates on parts and cheap milling of the PCBs. The easiest solution is the Lynx Express, which simply controls 16 120v AC channels. Closer to what you described, though, is the Lynx Freestyle, which drives 128 channels of control to 32 Lynx SSR4 boxes, which each hold 4 solid-state relays. There are also wireless modules, motor controllers, and a variety of other accessories.
Everything's managed by a normal computer. I know there's some software out there for Linux, but I don't know how well it'd work on a Raspberry Pi. Typically, a main control line runs through a USB interface to a control PC, which doesn't need to have particularly robust hardware. There are also some devices to play back a preprogrammed sequence. All the control system really needs to do is run the sequence.
That sequence is what takes most of the time. Effectively, you're choreographing a dance for a cast of thousands, but your dancers can't move, unless you have a few motors pushing things into the path of this analogy. If a show is too flash and high-tempo, it's hard for the viewers to appreciate the elements. Too slow, and the show gets boring. Too bright, and the neighborhood can't sleep. Too dim, and the show won't be visible. While sequencing, you also have to mind where your elements are, keep a few surprises for the audience, and have a bit of rhythm, as well. Of course, you have to do all of this enough to fill a show, anywhere from a few minutes to a full evening's set.
Without a soundtrack, though, the audience won't likely get the full effect of the show. Your carefully-choreographed masterpiece will just be a bunch of blinking lights. Some builders with small shows will just run outdoor speakers, but for wider coverage a radio transmitter will allow the audience to watch from the warm comfort of their cars. The legal ramifications of broadcasting Christmas music are often not discussed. Usually, the soundtrack is run from the same computer that controls the lights, as the control software will handle maintaining synchronization.
As for the music itself, many builders go with the classics, with a modern style. Trans-Siberian Orchestra and Mannheim Steamroller make modern arrangements from traditional tunes, practically perfect for this kind of production, as a blend of traditional decoration with a high-tech flair. Other builders throw in classic rock & roll, pop hits, and anything else for a more secular show. Some builders avoid Christmas altogether, running their theatrics at Halloween, where animatronics, special effects, and costuming share the stage for a literally fantastic experience.
Building light shows is a hobby that quickly turns into an obsession. There's always that one weak part in the sequence, or that really great element idea, or that song that is just so awesome. The community is supportive and creative, and the shows can grow according to any available budget. Fueled by the Internet's ability to connect people, builders have the support of a large community, but within any city there's few enough participants that shows are still novel and unique. For those who love theatrics, there are few better pastimes.
For those who love the holiday season, the shows are also a beloved spectacle. Builders often notice regulars who return to their shows night after night, year after year. The builder I worked with once received a card from a viewer who had come to his show several times, driving 45 minutes each way, just to watch. I've stood in the middle of the show with children who are amazed by every new element and will spend hours talking about their most-favoritest parts. That's what makes all the effort worth it... that moment where you realize that your show isn't just a show any more. It's a production; the product of your hard work, and on that dark night in the middle of winter, it is the most awesome thing in town.