Relativistic mass is just another way to characterize a massive object's momentum albeit in a way that can confuse everyone. Much like momentum I can always boost myself to a reference frame where that momentum is gone. As far as gravity is concerned, again, relativistic mass will be relevant to reference frame dependent effects (so I can't make black holes by boosting massive objects!!), worse still, as usual everything is more complicated in General Relativity and there are multiple ways to define mass and in some situations it might not even be a well defined quantity as mass can be exchanged with the gravitational field.

However at least in Special Relativity, I cannot do the same to get rid of an object's rest mass. The rest mass is the true quantity which all observers agree on. Relativistic mass came about because in the momentum equation,

p = m(rest)v/√(1-(v/c)^2)  = m(rest)v*gamma
p = m(rel.)v  

E(rest) = m(rest)c^2
E(rel.) = m(rel.)c^2

As we approach the speed of light, the momentum blows up, but the speed doesn't change much, thus the inertia from this reference frame is blowing up. In Newtonian physics we like the idea of equating inertia, acceleration and mass (via p = mv) and thus relativistic mass was born absorbing the gamma factor into the mass.

There isn't anything wrong with doing this, it's all about how you as a physicist wish to define the relationship between momentum and acceleration. The current educational consensus is to place the object in flatspace at rest (away from curvature) and write down the number in the low speed limit--then when we move onto more complicated physics, we carry all the coefficients and factors which modify that number all while keeping "rest mass" a constant property of the object. It's a human choice, we just need to be cognizant of what the math is telling us.

Here's a pro and con relativistic mass discussion by Philip Gibbs and Jim Carr,
http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html
http://www.phys.ncku.edu.tw/mirrors/physicsfaq_old/Relativity/BlackHoles/black_fast.html

Edit: These days, the only two masses I ever see explicitly written are rest mass and invariant mass. Here we need to be careful for in Special Relativity, m(invarient) != sum of m(rest). Motion (which cannot be translated away) contributes to the invariant mass and thus the relevant mass when you want to calculate gravitational effects. In this way, relativistic mass lives on in a subdued manner--by making your hot cup of tea more massive than your cold cup of tea.