I checked the numbers before posting this, so they should be very close to accurate. Based on what the Lufthansa website says, cm (in).
Carry-on -- 55 (21.65) x 40 (15.75) x 23 (9.06), 8 kg (17.64 lbs).
Personal item (briefcase example under "additional items..." toward the bottom of the webpage) -- 40 (15.75) x 30 (11.8) x 10 (3.94).
The Eagle Creek Load Warrior 20 International Carry-on I recently bought will work. Be sure to scroll down the page for a comparison of their sizes of this 2-wheeled bag, cm (in).
51.5 (20.25) x 31.5 (14) x 20 (8), 2.08 kg (4.5 lbs).
My new personal item, the Sea to Summit Ultra-Sil Sling Bag, fits their personal size requirements almost perfectly, too.
34.54 (13.6) x 29.97 (11.8) x 10.16 (4), 64 g (2.2 oz).
I got these bags for the light weight and for internal European flights. I haven't used either of them yet, but they should work, assuming that I don't stuff them so full that they bulge out.