What is an effective core workout routine that can be done without equipment other than dumbbells?Any type of exercise that requires a degree of balance will always force the transverse abdominis to contract in order to stabilize the body.
The most common exercises to engage the core are bridging type exercises. These can also be performed with dumbbells. Here are three good bridging exercises for the core you can do with dumbbells.
1. Plank with weight transfer: Get yourself into the regular plank position ( that's a press-up position but with elbows and forearms flush on the floor). Place the dumbbell on the floor at the side of you. Now holding your position reach out and grab the dumbbell. Transfer the dumbbell to the opposite hand and place the dumbbell on the floor at the opposite side of the starting position. Repeat. This should be performed for a period of time rather then reps. The slower, the tougher.
2. Core row: Get into the press-up position whilst holding the dumbbells. Arms should remain extended. Now row each dumbbell by pulling the dumbbell up to your rib cage. Lower the dumbbell slowly back to the floor. Now do the other arm. Again, focus on time rather than reps.
3. Rollout: This is a variation of the ab roller. Simple hold the end of the dumbbells making sure they have some weights on (I'm assuming you have those home-style dumbbells that you can attach weight to). Now push the dumbbell forward allowing the hips to follow (your knees should still be planted onto the floor). Now roll-out as far as possible keeping the back straight. Then pull yourself back to the starting position. Reps such as 10-12 is good (although well conditioned athletes will do much more).
Roll-outs are a tough exercise. The only problem with doing them with a dumbbell is they could make your hands quite sore (I've tried it myself). However, if you can overcome this with a good pair of gloves for example, it is a brilliant exercise for the core (personally I think it's the best of all core exercises).