The answer is simple, but you may not like it--offer more money. Depending on the market you are in, you may have to offer full price, a little more than full price, or perhaps substantially more than full price. And while that hurts, if you plan to keep the home for a long time and are in an area of rising prices, it may be your best option.
Think of it from the seller's perspective. In most cases, at close to the same price a seller would prefer a contract with no financing contingency, because it's one less thing that can go wrong. But most sellers also want to maximize what they get out of the house, so if your offer is substantially higher, and you can show you are well-qualified, some sellers may take a chance and accept your offer.
One other thing you are fighting is that cash offers usually allow quick settlement, which you might not be able to match. So again, you compete by offering a higher price.
Finally, one other big variable is inspection/repairs. It may help to arrange a pre-inspection, and if it's satisfactory make your offer not contingent on an inspection. That's a little risky, but not uncommon in really hot markets.
The bottom line is to think like a seller, figure out what would make you accept an offer over a full cash offer, and do that. In all probability, you may not get as good a deal as you would like, but that's the simple truth when there are a lot of cash buyers. You still may end up with a nice house which you enjoy and which appreciates over the long haul, so the extra $10-50K you may spend will end up being worth it.