There are a few different reasons you might be having a harder time in 3s:
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The meta is completely different than summoners rift. Certain champions, particularly those that are strong fighting in tight spaces, are much stronger. 
& 
are all examples of this.
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Your MMR/ELO from other modes doesn't carry over. In low ELO 3s you can battle people from across the spectrum of 5s ratings. A player who is iron/bronze/silver in 3s can easily be platinum+ in 5s, with mechanics and matchup knowledge commensurate to that rating. This can create my massive mismatches in mechanical skill, since it's possible for you to have a player on your team that is bronze in both 5s and 3s attempting to lane against a player that is bronze in 3s but is diamond in 5s. This can be somewhat counterbalanced by having good 3s specific macro knowledge.
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Compositional strength matters more in 3s. Since each member of your team is theoretically 33% of your team's overall strength, one person having a bad mismatch matters all the more. Additionally since each team member matters so much, you need a cohesive team composition. The most common compositions in threes are jungle, funnel/support, and team fighting.
Jungle comps typically feature a bruiser/fighter in the top lane, a strong ganking jungler, and a mage/marksman in the bot lane. The aim of this type of composition is for the jungler to focus on getting the laners ahead early so that in turn the laners can help invade the opposing jungle and leave 1 to 2 members of the opposing team so weak that they are functionally down numbers.
Funnel/Support comps feature a tank support in the top lane, a hyper carry jungler, and a strong laning mage/enchantress bot lane. These types of compositions attempt to funnel extra resources from the top lane into the jungler so that the jungler can hard carry. This is done by the top laner roaming to help the jungler clear camps more quickly, as well as using a support item and giving over the majority of their cs to the jungler.
Team fighting compositions feature scaling carry picks top and bot paired with a strong tank jungler. These compositions attempt to use the faster rate of gold/experience gain on the 3s map to "scale for late game" more quickly. The jungler protects and frontlines for the carries in the early to mid game so that they can outscale in front to back team fights at objectives in the mid to late game.
