  We substantiate by calculations that the recently discovered
superconductivity below 4 K in 3% boron-doped diamond is caused by
electron-phonon coupling of the same type as in MgB$_2$, albeit in 3
dimensions. Holes at the top of the zone-centered, degenerate $\sigma$-bonding
valence band couple strongly to the optical bond-stretching modes. The increase
from 2 to 3 dimensions reduces the mode-softening crucial for $T_{c}$ reaching
40 K in MgB$_{2}.$ Even if diamond had the same \emph{bare} coupling constant
as MgB$_{2},$ which could be achieved with 10% doping, $T_{c}$ would only be 25
K. Superconductivity above 1 K in Si (Ge) requires hole-doping beyond 5% (10%).
