Groveton sports a covered bridge right next to route 3. There is an interesting railroad engine next to it.
Further north just off route 3 there is a covered bridge to Vermont. You can just see it on the left in Columbia.
Pittsburg has 3 covered bridges.
There are two covered bridges in Lancaster. One is near the middle of town, and the other on route 135, south of town. To get to each of them, find the Town Hall on Main St. For the closest, go up Mechanic street beside the Town Hall. You'll see the covered bridge as soon as you start up the street. The other one is on the road that starts across Main Street from the Town Hall. The road is marked Elm Street, and State Route 135. The Mt. Orne covered bridge is about 4 miles down that road, crossing the Connecticut River to Lunenberg Vermont. It is quite long, and at the other side is a spring with delicious, cold water.
If you pay close attention, about a mile south of the hamlet of Lancaster, you'll see a sign for Lufkin's Auto Body. If you go out that dirt road a short distance, you'll encounter the Lufkin's herd of buffalo! Early in the morning is often a good time to see them as he is feeding them then. Do NOT cross the fence! They can run faster than you!
One of the most photographed, painted, and beautiful covered bridges in the northeast is about 15 miles north of Lancaster. Take State Route 3 north to Groveton, then State Route 110 towards Berlin. About 5 miles down the road, you'll see the bridge on the left in Stark.
You can also look here at White Mountains Covered Bridges - A complete list of covered bridges in New Hampshire.