1996 royal star, left rear valve seems to be sticking or something, getting rough idle with lots of backfire popping from that pipe, and motor is missing badly during takeoff and downshift idling. anybody have any ideas?

what color are the are the spark plugs does it have aftermarket pipes or goodies for the motor

Backfiring is a sign of an air leak. Somehow air is being allowed into the exhaust system. If you or a previous owner has drilled out the baffles to increase sound, this could be the cause. A burned valve could contribute but is not likely unless the engine has been severely overheated. A burned valve will also cause poor performance and it can pop upon acceleration which was not mentioned in your original question so I doubt it is a valve causing the problem. Plus - these engines are pretty reliable and robust. I bet on the air leak.

Scott

replce your
valve seal

The firing order on the V-4 engine in Royal Stars from 1996 up is 1-3-2-4, number 1 being the left (shift side) rear cylinder, 2 the right rear, 3 the left front, and 4 the right front.

You have either a stuck or damaged valve due to overheating. Straight pipes are bad for Yamaha's because they remove all back pressure, causing the air to flow too fast through your Carbs and sometimes dislodging O-Rings. A performance carb will eliminate this problem and provide better air-flow. You can try some performance carbs by Mikuni or Kei-Hinn, but you will need to deal with that valve first. If you are lucky, it may just be sticky, but if it's cracked you can do a compression test to see how bad, or even if it is.

valve sticking,, ring stuck

Exhaust 'pops' is generally caused by unburnt fuel igniting in the exhaust, air leaks can cause this too.
When you do any carb work its always best practice to check the carb balance and idle mixture settings.
What colour is the spark plug? http://www.dansmc.com/Spark_Plugs/Spark_Plugs_catalog.html

YAMAHA ROYAL STAR Engine Type: 1294cc liquid cooled 70 degree DOHC V-four, four valves per cylinder
Bore and Stroke: 79 x 66mm
Compression Ratio: 10.1:1
Carburetor: 4 x 28mm Mikuni CV
Transmission: 5-speed
Wheelbase: 66.7 in.
Dry Weight: 701 lbs.
Tires: Front; 150/80V16 Rear; 150/90V15
Forks: 43mm Kayaba
Rear Suspension: Kayaba mono-shock with adjustable preload
Brakes: 298mm dual front discs with dual-piston calipers; 320mm rear disc with dual-piston calipers
Seat Height: 28.3 in.
Fuel Capacity: 4.8 gallons

sounds like you left something out. check to see if you plugged in all the sensors and plugs, then check for any vacuum lines that were not replaced or may have been broken