Some onboard sound is noisy through the analog outputs, although I guess it's only really noticeable in headphones. For normal PC speaker e-mail notifications and whatnot, it doesn't matter.
But luckily, most motherboards have S/PDIF outputs via coax and/or TOSLINK that allow you to connect to an external DAC (like the ~$40 FiiO D3) or a reciever with digital inputs. Some all-in-one PCs and laptops (all Macbooks IIRC) have a combined headphone output and mini-TOSLINK jack, but even if they don't, you can do digital audio over USB, Firewire or Thunderbolt for not a lot of money. HDMI or DisplayPort can also transport sound, often a DAC in the monitor allows analog stereo output based on that.
The point is to remove the noise-sensitive analog parts of the signal chain from inside the PC, using a digital connection. For stereo sound, a dedicated sound card is completely overkill, when a $40 DAC and a $5 TOSLINK or coax cable will provide an even lower noise floor in real-life situations.
For surround sound applications, dedicated sound cards may still have a place, but using a digital connection to an AV receiver and letting it handle all the processing is a much better solution and integrates better with TVs, consoles etc.