In the state of Texas, you aren't allowed to operate a self-built moped on their public roadways. A moped must be on a certified manufacturers list which is updated every 90 days in the state of Texas.
Until/if laws ever get changed; here's a video of something you can legally operate in Texas. A self-built motor-assisted scooter. You can get manufactured 49cc seated scooters like this and probably won't be harassed as long as you obey all traffic laws, don't exceed 35 mph, and don't ride on streets with a posted speed limit higher than 35 mph. However one could probably get away using a bicycle as long as the engine wasn't over 40cc, you obey all traffic laws, you don't go over 35 mph, and stay off streets with a speed limit posted higher than 35 mph. If going any of these routes get some kind of speedometer device.
In the state of Texas, a motor-assisted scooter isn't allowed to operate on any public roadway which has a posted speed limit over 35 mph.