"Do I really have to water when it's been raining so much?"
That's the short answer.
Now here's the long answer;
A soft, gentle rain is very beneficial to your garden but it doesn't take the place of a thorough watering done by....you. A few minutes in a torrential down pour may soak you to the skin but it will certainly not be enough to keep your gardens alive.
Let's talk about in-the-ground gardens around your house first.
Now that we're in the summer months your plants have had time to establish a nice deep root system. A short heavy rainfall will moisten the top of the soil but deep below where your plants' roots are, may still be dry. Stick your hand or trowel down 5 or 6" and check the moistness down there. Dry?? Drag the hose out and get watering.
What about hanging baskets and containers?
First and foremost, make sure your containers aren't getting washed out if they are directly under an overflowing eavestrough or downspout. If so, move them and tip out as much excess water as possible. The only source of nutrients that your containers have is from the fertilizers that you've fed them and if your plants are getting soaked by the rain then all the goodness that was in the soil has been leached. Make sure your containers have proper drainage, apply fertilizer and allow your pots to dry out a bit.