A practical way to check if a matrix is positive-definite

These matrices are called (strictly) diagonally dominant. The standard way to show they are positive definite is with the Gershgorin Circle Theorem. Your weaker condition does not give positive definiteness; a counterexample is $ \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 1 \end{matrix} \right] $.

Here is another way to see this. Define $R_i = A_{ii} - \sum_{j\neq i} \lVert A_{ij} \rVert$. Your condition is that $R_i>0$ for all $i$.

Let $s_{ij} = \frac{A_{ij}}{\lVert A_{ij}\rVert}$ be the sign of $A_{ij}$. Then you can check algebraically (just match coefficients) that for all $x\in\mathbb{R}^n$ \[x^T A x = \sum_{i=1}^n R_ix_i^2 + \sum_{i=1}^n \sum_{j>i} \lVert A_{ij}\rVert (x_i + s_{ij} x_j)^2. \] Since squares are nonnegative and the $R_i$ are assumed positive, all summands are nonnegative for all $x\in\mathbb{R}^n$. Furthermore, if $x\neq 0$ then $x_i\neq 0$ for some $i$, so $x^TAx\geq R_ix_i^2>0$. Therefore $A$ is positive definite.

This expression for $x^TAx$ can be alternatively viewed as expressing $A$ as a weighted sum $A = \sum_k c_k v_kv_k^T$, where each $c_k>0$ and each $v_k\in\mathbb{R}^n$ is a vector with support (number of nonzero entries) at most two, each of which is $\pm 1$. But $v_kv_k^T$ is always positive semidefinite for $v_k\in\mathbb{R}^n$ and positive combinations of positive semidefinite matrices are positive semidefinite. Since each $R_i>0$, we can decrease some of the $c_k$ slightly (those which correspond to the $v_k$ which are standard unit vectors) and instead write $A = cI + \sum_k c_k v_kv_k^T$ for $c>0$, which shows that $A$ is in fact positive definite.

One nice thing about this proof: Every positive definite (or semidefinite) matrix can be written as a positive combination of matrices $vv^T$, but this proof shows that for diagonally dominant matrices we can take all the $v$ to have support at most $2$. This gives some intuition for why "most" positive definite matrices are not diagonally dominant. For example if $v$ is any vector of support size at least three then for small enough $c$, $cI + vv^T$ is positive definite but not diagonally dominant.

I cannot imagine this is difficult.

Before continuing, let me add the caution that a symmetric matrix can violate your rules and still be positive definite, give me a minute to check the eigenvalues

$$ A_3 \; = \; \left( \begin{array}{rrr} 3 & 2 & 0 \\ 2 & 3 & 2 \\ 0 & 2 & 3 \end{array} \right) , $$

Yes, that is fine, eigenvalues are $3 - \sqrt 8, \; 3, \; 3 + \sqrt 8$

A proof is given here http://planetmath.org/?op=getobj&from=objects&id=7483 as a consequence of Gershgorin's circle theorem. For additional information, see http://en.wikipedia.org/wiki/Diagonally_dominant_matrix and http://mathworld.wolfram.com/DiagonallyDominantMatrix.html or just Google "diagonally dominant symmetric"