We have the topography we do largely because of erosion rates, size of plates, and transportation rates. To generate a "rougher" planet you need to dial down erosion rates, dial down size of plates, or dial up transportation. (Or any combination of the three.)
1. Erosion rates depend largely on overall chemical composition and on hydrology. Briefly, to generate lower erosion rates you want your rocks to be more mafic (containing heavier elements) and less felsic (containing lighter elements). This is a matter of the composition of and position in the planetary disc that formed your system, so you'll want your planet to have developed in a region around its star a little closer than ours1.
As for hydrology, you want a little less vigorous of a water cycle than ours. You could make the planet contain significantly less water: say, 20% coverage rather than 80% coverage. Alternatively, you could have the same coverage but slow down the water cycle by slowing down evaporation. (This can be done either by cooling the planet or by slowing down its rotational rate2.)
2. Plasticity of plates and other factors will determine how large your tectonic plates are. You might think it unfortunate that all of the factors in (1) that make your rocks harder and less prone to erosion also make your plates harder, but realize that makes plates more brittle, too. To break them up into generally-smaller fragments we just need to make sure the planet's getting kneaded vigorously: ditch any thoughts of a moon and give yourself a true binary planet system.
3. Transportation of plates arises from convection cells in the mantle, so we want to make sure those cells are more vigorous. Unfortunately, the chemical factors in (1) that make rocks harder also make magma more less viscous, so we've got to invigorate convection cells in another way. We can either crank up the (core) heat or spin up the differential rotation rates between core and crust. Luckily, if we're creating a planet closer to its parent than we are, then it was likely created in a region of the planetary disc more prone to energetic vortices, and it's not unreasonable for it to have a fast-spinning core. Couple that with a slow binary partner and we can imagine a slow crust and fast core creating vigorous--and relatively small--convection cells. Perhaps that'll lead to smaller tectonic plates.3
In summary, you've got chemical composition, water coverage, surface temperature, surface rotation rate, size of orbital partner, core rotation rate, core temperature which will all impact eventual geomorphology. I hope above to have explained enough of how each works and how they might interact that you can make your choices.
There may be many more things I'm not thinking of or even aware of. Happy hunting!
1: The relative abundance of heavy/light elements vary as one approaches the central star in a planetary disc the same way the relative abundance of heavy/light gases varies with altitude. And it only takes a difference of a few % silicon abundance to radically change rocks' properties!
2: slowing down the rotation will diminish prevailing wind bands, but will increase thermal differentials at the terminator. Effectively, your trade winds and polar/equatorial easterlies will die off (less evaporation), but pre-dawn and dusk sea-breeze and shore-breeze will get stronger (more evaporation). Since I've got no idea which would effect evaporation more, I suggest you stay with our rotation and push on some of the other levers.
3: Unfortunately, you've probably just done big things to your magnetic field: your planet has a larger metal (core) due to its original abundance-profile, and it's spinning faster. YMMV.4
4: "Your Magnetism May Vary"