To get all three across the river, you can follow these steps:

1. Take the monkey across the river, leaving the tiger and the banana on the original side.

2. Leave the monkey on the far side of the river, and return to the original side of the river.

3. Take the tiger across the river, leaving the banana on the original side.

4. Leave the tiger on the far side of the river, and pick up the monkey.

5. Take the monkey back across the river, and leave it on the original side.

6. Finally, take the banana across the river and leave it with the tiger.

This way, each animal is transported across the river without being left alone with its natural predator, and all three animals and the banana end up on the other side of the river.