I have used the randomForest package in R several times and there were some functions to measure the variable importance such as importance() and varImpPlot(). As far as I know varImpPlot visualizes the the importance of each predictor with respect to variables' contribution in the decrease of error measures (e.g mean squared error for regression, Gini index for classification etc.)
What I usually do to measure variable importance in a really simple way is, I estimate linear and lasso regressions and then see how much the coefficients were shrunk.
library(MASS)
library(randomForest)
library(glmnet)
data(Boston)
# Random forest (based on the lab example from the book "An Introduction to Statistical Learning")
rf.boston <- randomForest(medv ~., data = Boston, mtry = 13, ntree = 25, importance = TRUE)
importance(rf.boston)
%IncMSE IncNodePurity
crim 2.80848132 1340.74773
zn 2.12135233 34.74243
indus 1.49676063 270.12398
chas 0.06577971 31.42226
nox 7.42985606 1381.58615
rm 13.41323143 18128.73241
age 6.28896854 487.95644
dis 7.08361676 2621.61526
rad 1.71445398 128.88846
tax 6.48150760 557.31305
ptratio 5.24860362 660.97934
black 2.00139088 562.16876
lstat 9.26159315 16553.01919
varImpPlot(rf.boston)

# Linear model
lm.boston <- lm(formula = medv~., data = Boston)
# Lasso
optim.lambda <- cv.glmnet(x = as.matrix(Boston[, -14]), y = as.vector(Boston[, 14]))$lambda.1se
lasso.boston<- glmnet(x = as.matrix(Boston[, -14]), y = as.vector(Boston[, 14]),
lambda = optim.lambda)
sum.abs <- abs(coef(lasso.boston)[-1])/ abs(coef(lm.boston)[-1])
sum.abs <- sum.abs[order(sum.abs, decreasing = F)]
barplot(sum.abs, horiz = T, col = "red", las=2)

par(mfrow = c(2,1))
rf.boston <- randomForest(medv ~., data = Boston, mtry = 13, ntree = 25, importance = FALSE)
varImpPlot(rf.boston, main = "Variable importance (Random forest)")
barplot(sum.abs, horiz = T, col = "red", las=2, main = "Variable importance (Lasso)")
And a quick comparison for both:

As far as I understand, Lasso approach could be a bit problematic when predictors are correlated. You can see that rm variable has a larger lasso coefficient than the linear one.