Here is a possible suggestion how such an approach might work. The mechanism follows from rather general considerations and does not require quantum mechanics. So I'll use a more general mathematical notation.
Imagine I want to know the time evolution $f(x,t)$ when I only have knowledge of the initial condition $f(x,0)$. The assumption is that there is some dynamics that uniquely fixes $f(x,t)$, given $f(x,0)$. One can express the dynamics by some linear operator (linear equation of motion).
One way, as you pointed out, is to expand the solution in terms of the eigen-functions (let's briefly review it)
$$ f(x,t) = \int F(\omega) \phi(x,\omega,t)\ d\omega . $$
Here $\phi(x,\omega,t)$ are the eigen-functions and they are indexed by $\omega$ (associated with the energy). These eigen-functions are also orthogonal in the sense that
$$ \int \phi(x,\omega,t) \phi^*(x,\omega',t)\ dx = \delta(\omega-\omega'). $$
This now allows us to obtain the spectrum for $f(x,0)$ at $t=0$ using
$$ F(\omega) = \int f(x,0) \phi^*(x,\omega,0)\ dx . $$
Then we can substitute $F(\omega)$ into the original expansion to get a general expression for $f(x,t)$ that is valid for all time.
Now for the alternative approach. Imagine we do the substitution, just mentioned and then change the order of integration
$$ f(x,t) = \int f(x',0) \int \phi^*(x',\omega,0) \phi(x,\omega,t)\ d\omega \ dx'. $$
The inner integral now represents a Green function or propagator for the process
$$ K(x,x',t) = \int \phi^*(x',\omega,0) \phi(x,\omega,t)\ d\omega. $$
If the dynamics is translation invariant in $x$, we'll get
$$ K(x-x',t) = \int \phi^*(x',\omega,0) \phi(x,\omega,t)\ d\omega. $$
If we substitute this back we obtain a convolution integral
$$ f(x,t) = \int f(x',0) K(x-x',t)\ dx'. $$
So this provides an alternative approach. However, one needs to know the Green function, or alternatively derive it from knowledge of the eigen-functions.
