What the OP wants is perfectly possible. I'm typing this on an Ubuntu 12.04 box running the most recent Catalyst driver, and connected to three 1920x1080 monitors. Two are DVI, one is via a DisplayPort->DVI adapter. Video card is an older Radeon 6950. It works, more or less without issue, for what I do: coding in Eclipse, browsing the Internet, etc.
Using the open-source driver works for triple monitors, but the power management is not up to snuff in the open-source driver, and the fan on the video card gets annoyingly loud after a few minutes. This is the only reason I run the closed-source driver. Strangely video playback is smoother with the open-source driver in the triple monitor scenario.
Contrary to popular myths, you do not have to edit a config file for either closed or open-source drivers to enable magical triple monitor goodness. Both were able to detect and orient the monitors using either the Ubuntu monitor control panel or the Catalyst Control Center.
Things that don't work as well: video playback and 3D games. Video will get choppy full-screen if tear-free mode is enabled, and the tearing is intolerable when it's not. Likewise, performance for 3D games across 5760x1080 is iffy. I have a laptop for gaming and an HTPC for the video stuff, so it's not a deal-breaker for me. The OP did not specify what kind of engineering he/she does (circuit design? CAD? software?), so the 3D performance may well be an issue depending on the tools being used.
I have tried the Nvidia route several times, but always came away frustrated. AMD cards Just Worked for this application. Google 'Linus Torvalds middle finger' for a more complete technical discussion of why this is.
Getting a reliable triple monitor setup on Windows or Mac is much easier than in Linux, but most that experience can be chalked up to X. In theory, Wayland or Mir will handle this much better, but no stable distro uses them by default, and none of the high-level toolkits have mature support for it.