[Warning: this advice is based on the experience of one player and not anything scientific or precise. High-ranked Rolling Pins players are encouraged to edit this!]

Try to knock down as few pins as possible early on in the frame so that you have more flexibility later on.

Knocking down too many odd-numbered pins early on can hurt your flexibility. You can't avoid knocking down an odd-numbered pin whenever you roll an odd number (except by using a bowling ball), but once they're all gone, only even numbers remain as good rolls, meaning every roll has at least a 50% chance of being bad.

Try to knock down the back pins (7, 8, 9, and 10) as quickly as possible, since they can only be hit with 2 dice. Once those are all gone, you have the option of rolling only 1 die. Rolling 2 dice is still more efficient if you have a lot of the front pins standing, but when you need low rolls, a single die is the way to go.

Much of the strategy in this game boils down to when to use the bowling ball power-up to increase or decrease your roll. The scoring system of bowling heavily rewards strikes, but especially rewards getting multiple strikes in a row (three strikes in a row is called a "turkey", and is worth at least 60 points despite only knocking down 30 pins). Therefore, you should orient yourself towards maximizing your chances of getting a turkey, or an even longer streak of strikes, when deciding when to use balls and when to save them. Early on, you should be extremely reluctant to use balls unless it gets you a guaranteed strike (for example, changing a roll of 7 to a roll of 8 when the 8 pin is the last one you need); even then, you might choose to wait if it's very early (say, frame 1 or 2) and you aren't well-equipped to turn the strike into a turkey. In the mid-game, especially when you're on the verge of a turkey, consider using balls to maintain flexibility and hit difficult pins (for example, changing a roll of 9 to a roll of 10 when your 1, 3, 5, and 10 pins remain, as knocking down 10 alone leaves you more flexible than knocking down 1+3+5). Of course, on the final frame, there is no point to saving balls anymore, so use them liberally.