There are some older 12" panels up to about 800x600 that have a parallel interface - these are really easy to drive via LVDS.
There is a parallel to LVDS add-on board for the Beagleboard, which may be a ready-made solution
1Kx768 LVDS direct from a FPGA like the Spartan6 that has serialiser support is probably not that hard - I'm sure makers of FPGAs with this stuff will have examples and appnotes
Bear in mind that at this sort of resolution, getting enougg memory with sufficient starts to become nontrivial unless you're doing simple non memory-bitmapped stuff