I mean something where just participation in the decentralized network gives you coins somehow and you cannot do anything beyond (excluding more participation, of course, by having multiple PCs and IPs, for example) to increase the rate of achieving those coins. At the moment I have completely no motivation to leave my bitcoin or litecoin client on, because it doesn't gives me anything back for using me as a node in this network of hardcore miners. I'd rather find some ways to use it occasionally only when I need to do some transactions (few times a month, for example). I'm newbie in this topic, but doesn't this harm the network, when lot of people become using it this way?
On Bitcoin and Red Balloons suggests a method to incentivize network nodes, however no cryptocurrency implements it.
Cuckoo Cycle is close. It is a memory-hard proof of work system - substituting random memory access for CPU power.
The up and coming Safecoin will reward users for donating their bandwith and harddrive space, so they'd be "mining" by just leaving their computers on and connected, being compensated in safecoins.
To a lesser extent, the way forging in NXT works you also need to leave your computer turned on and connected the whole time so as to be taken into account for the raffle of who generates the next coin, though that doesn't ensure you'll receive any NXTcoins: since it's a 100% Proof-of-Stake coin the more coins you have the higher your probabilities of forging the next block.