The problem here is that Mathematica doesn't recognize `{x,y,z}` as some kind of a vector object that should be treated as grouped together; instead, it substitutes in three independent variables, and probably starts integrating them one by one. The result is a very complicated integral.

If you do the coordinate transformation yourself, you can reproduce the result. Simply use $\mathrm dp = \mathrm d\|p\|\|p\|^2\mathrm d\vartheta\sin(\vartheta)$ to transform to spherical, the resulting integral can be calculated:

    Assuming[pAbs >= 0 && m > 0 && r > 0,
        (1/(2*Pi)^3)*Integrate[ (* phi integral *)
            Integrate[ (* |p| integral *)
                Integrate[ (* theta integral *)
                    (Exp[I*r*pAbs*Cos[pTheta]]/(pAbs^2 + m^2))*pAbs^2*Sin[pTheta],
                    {pTheta, 0, Pi}
                ],
                {pAbs, 0, Infinity}
            ],
            {pPhi, 0, 2*Pi}
        ]
    ]

> $\displaystyle\frac{e^{-mr}}{4\pi r}$