This really comes as a surprise to most hunters, but Free Hunt is amazing.
Don't confuse Free Hunt with the Moga Village Free Hunt, where you only gather resources. By Free Hunt I mean hunting monsters that 'invade' your quest. The most famous invader is the Deviljho you may have encountered when hunting a Qurupeco. You can tell you have a chance at free hunting a monster when the quest description includes the 'danger' icon (i.e. unstable environment). The monsters can show up at random times and may be selected from a pool of potential monsters. You can find more info about what monsters can invade what quests in Ping's MH3G Dex.
But what makes Free Hunt so good? Well, consider this. When you Free Hunt an invading monster, you can break parts, you can hunt or capture it and sometimes you can cut tails – all these things net you the normal rewards. But what makes Free Hunt not just good, but amazing is the Free Hunt Bonus screen after you finish your quest. You get a whole row of quest rewards, where you have up to 8 goes as the Free Hunt rewards pool – and what an amazing pool it is. Free Hunt is a hidden gem (that often drops many, many gems) in MH3U. To see for yourself, have a browse of my Carves & Drops guide on GameFAQs.