In the hopes of helping anyone similarly situated save time by staying close to a known good configuration, I am going to describe my present system configuration and also try to list what did not work.
Monitors:
Seiki se39uy04 3840x2160 @ 30Hz HDMI
Hewlett-Packard HP LP3065 2560x1600 @ 60Hz dual link DVI (via DisplayPort)
Motherboard: MSI b85m-g43
Processor: Intel i7-4770k (relevant because it contains the video silicon)
Memory: 2x 8GB pc3-1600 unregistered DIMM's
DisplayPort adapter: (Dell?) BizLink (but does not pass EDID video timing information, which is a big pain)
Software: Linux-3.13.0-64bit kernel (sometimes 3.14-rc4-64bit), X.org X server 1.12.2.901, xf86-video-intel 2.99.910 (using kernel-based video timing mode setting).
Hardware that did not fully work:
Video cards that did not work (all of them successfully do 3840x2160 @ ~15Hz, but not 30Hz):
Asus Radeon 6450 - Dual Link DVI OK, but cannot drive 4k @ 30Hz HDMI (probably a hardware limitation)
XFX Radeon r7-240 (the fanless version of this card) - No dual link DVI, cannot drive 4k @ 30Hz HDMI (HDMI problem is probably a Linux kernel bug)
XFX Radeon r7-250a-zlh4 - same as r7-240, even though TigerDirect describes it as having a "1 x Dual-Link DVI-D."
The XFX r7-240 and 250 cards supposedly have a chip that only has two digital links: one for HDMI and one for single link DVI. So, I don't think the that lack of dual link DVI is a driver bug.
DisplayPort adapters that did not fully work with my HP LP3065 2560x1600 dual link DVI display:
Okeba, VisionTek 900639 - these two failed in slightly different ways. Both were able to do 1280x800 single link, both were advertised as supporting dual link
(Dell?) BizLink - I can drive 2560x1600, but had to manually specify all of the video timings (using "cvt -r 2560 1600", "xrandr --newmode...", "xrandr --addmode...", etc.), because for some reason I not get the EDID video timings advertised by the display. Fortunately, the "cvt -r" timings exactly match the EDID timings from the monitor when I dump the information by connecting it from one of the other DisplayPort adapters. Using timings from cvt without "-r" results in my HP display blinking a few black frames every few minutes; with "-r" I have not seen any such blinks.
Cables:
Using the blue "UHD" cable that came with the Seiki resulted in the video going black for a second every now and then. Using an HDMI cable that I think I bought for $2 several years ago seems to work fine. Seiki's blue cable is very hard to bend. Perhaps I bent it at some senstive point near a connector in the course of connecting everything.
Other observations about the Seiki display:
I have not been able to get the Seiki to do a firmware upgrade. Maybe it is already running the latest and senses this. The firmware appears to be a Linux distribution. I read online that the Linux distribution is "Darwin Linux", that it may be possible to ssh into it, and that a chip in the display has a built-in ethernet port. Some people on that forum were talking about trying to reprogram an FPGA in the display to do 60Hz over two HDMI cables, but I wouldn't hold my breath for any of those hardware hacking plans to become reality.
Anyhow, I hope these notes might save somone considering 4k right now from a mistake or two.
In the preceeding notes, I am not trying to argue one way or the other about who should be an early adopter of this. Perhaps I'll try to make a separate posting describing the parts of my impressions so far that I think might help others decide. For now, I'll just say I'm happy with the result of this upgrade, in spite of how much work it has taken, but that has a lot to do with my particular situation.