Linxi in itself is not an interesting town. However, you will see very few tourists around here. There is a nice mountaineous area about 25 km west of Linxi, along main road number 303, at a place called Reshui. You get there by taxi (50 RMB) or by bus or mianbaoche for about 5 to 10 RMB. There is a train service, but the trains run too late in the evening.
The mountains south and west of Reshui are about 1500 meters high, sometimes like hills, to the west partially quite rocky. People live like in ancient times, but are very friendly. Because of some years where many foreigners visited this area because of steam locomotive hauled trains, people are used to see foreigners. At Reshui, there are lots of hotels, prices from 100 to 250 RMB per night. One of the best is the Jitong Railway hotel (Jitong tielu binguan). There is no problem in getting place, if you do not come in summer.
From there you may walk small pathes in the mountains, through villages. If you walk into one of the schools, children and teachers will welcome you.
If you want some real hike, take a taxi west from Reshui, about 20 kilometers over the Dabaliang pass. Then scramble up the HaDaShan mountain or the mountains behind the village of Tongxin (near a large curved railway viaduct). (main road 303)You find maps of the area on my web page.