These alternate routes, while not always faster than the normal route, may often give you more items than you'd normally get!
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Luigi Circuit: After you go off the first speed boost jump after the starting gate, if you hang a right there is a little path behind the chain chomp post. Before the second U-turn, there is another alternate route behind some pipes. I found it's not much use. (These for Luigi Circuit only work in 100cc, 150cc.)
Peach Beach: Before the first item row, there is a left turn that leads you into a pipe that shoots you out into a double item. Unfortunately, it does not help you any. Before the finish line, instead of going straight, turn left and go around the fountain. There are 2 double items, but it's longer to go that way.
Mushroom Bridge: Right after the starting line there is a ramp to the sharp right. Take it, go through a pipe, and you are shot out to receive a double item. This one is also longer than just going on the track. Also, after the first tunnel, there is a road to the left. Take it to avoid all the cars on this stretch of road and get an item.
Mario Circuit: In Mirror mode, use a mushroom to go behind the chain chomp through the grass. If you can do it right, it removes a few secs from your time. I found it best to use a light kart.