I don't agree that March - May would be the wrong time of year for a roundabout LA-SLC trip. Actually, it's probably the best time of year if you'll be spending most of your time in lower elevations. SteveB's comment about the 5,000 foot mark is certainly off - I lived most my life at 5,000 ft in the southwest, and in mid-April, that's perfect 70 degree weather, not anywhere near the rain/snow line that time of year.
Unless you're planning to cross the mountains right away, you'll likely have good weather. Depends greatly on your route, but March is perfect in much of Southern California, Arizona, and Southern Utah.
Most likely the coldest part of your trip would be around the Grand Canyon/Flagstaff areas, and if hit that area in late April, you can expect highs near 60 F and lows about 35 F - not really too bad for bike touring.
Last edited by DCycle; 12-03-17 at 07:44 PM.