The reduced power and reduced product constructions are generalizations of the ultrapower construction that use filters instead of ultrafilters, and you can construct a filter from a measure (and hence a reduced power) simply by taking the collection of all sets whose complement is measure 0. See the book "A Course in Universal Algebra" at http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf for the definition of a reduced power.

There is a weak version of Łoś's theorem that holds for reduced powers known as the Feferman-Vaught theorem, but it is a bit messier for reduced powers than for ultrapowers. This result basically gives a method of determining the truth value of a sentence in a reduced power from the truth value of different sentences in the factors of the reduced power, and this result also holds for generalizations of the reduced power construction such as limit reduced powers, reduced products, and Boolean ultraproducts. The Feferman-Vaught theorem can be found in standard texts on model theory such as Hodges and the text by Chang and Keisler. Let me now state the Feferman-Vaught theorem.

Given any first order sentence $\phi(x_{1},...,x_{n})$ in some first order language $\mathcal{L}$, the algorithm finds a sequence $(\sigma,\theta_{1},...,\theta_{m})$ of formulas such that $\sigma(z_{1},...,z_{m})$ is a formula in the language of Boolean algebras, and each formula $\theta_{i}$ has at most the variables $x_{1},...,x_{n}$ free and where we have the following:

Assume $I$ is an index set, $Z$ is a filter on the set $I$, and $\mathcal{A}_{i}$ is an $\mathcal{L}$-structure for $i\in I$. Assume also that $f_{1},...,f_{n}\in\prod_{i\in I}A_{i}$, and given $1\leq j\leq m$, let $R_{j}=\{i\in I|\mathcal{A}_{i}\models\theta_{j}(f_{1}(i),...,f_{n}(i))\}$. Then we have $$\prod_{i\in I}\mathcal{A}_{i}/Z\models\phi(f_{1}/Z,...,f_{n}/Z)$$ if and only if
$$P(I)/Z\models\sigma(R_{1}/Z,...,R_{m}/Z).$$

I personally found the above result to be very useful since I was able to determine the sentence algebras and the elementary classes of a certain variety using a version of the above result. Of course, the above result is not as powerful as Łoś's theorem since it does not automatically give you a non-standard model of a structure like the real numbers.