Inspired by SensibleDesign's multi-color Lulzbot logo, I decided to create one that could be printed without a multi-material system. I've attached files for all my attempts.
Lulzbot logo (3 layers).stl : this one is for a single extruder with two manual filament changes. Print 0 to 2mm in white, switch and print 2 to 3mm in green, switch and print 3 to 4mm in black. I wasn't entirely happy with this one, however, because the center white parts are hard to see when the green and black layers are printed; the 2mm recess is just too far.
Lulzbot logo (4 layers).stl : like the above, but with an extra white layer on top for the center bits. Intended for 0.2mm layer height, with first layer at 0.3mm. Print 0 to 1.1mm in white, switch and print 1.1 to 1.7 in green, switch and print 1.7 to 2.3 in black, then switch again and print 2.3 to 2.9 in white.
Lulzbot logo (DE ...).stl : for printing with dual extruder with a single filament change. Combine all 4 files in your slicer, and set it so green and black are printed by one extruder, white by the other. Print 0 to 1.1 in green and white, switch and print 1.1 to 1.9 in black and white.
For the filament changes, modify your slicer's gcode to insert a M600 command or use this snippet.
Fusion 360 files are attached in case you want to customize the logo plate further.