Does it make sense to study plots of residuals with respect to the dependent variable? I would like to know whether it makes sense to study the plots of residuals with respect to the dependent variable when I've got a univariate regression. If it makes sense, what does a strong, linear, growing correlation between residuals (on the y-axis) and the estimated values of the dependent variable (on the x-axis) mean? 

 A: Assuming the estimated model is correctly specified...
Let's denote $P_X=X(X'X)^{-1}X'$, the matrix $P_X$ is a projection matrix, so $P_X^2=P_X$ and $P_X'=P_X$.
$Cov(\hat{Y},\hat{e})=Cov(P_XY,(I-P_X)Y)=P_XCov(Y,Y)(I-P_X)'=\sigma^2P_X(I-P_X)=0$.
So the scatter-plot of residuals against predicted dependent variable should show no correlation.
But!
$Cov(Y,\hat{e})=Cov(Y,(I-P_X)Y)=Cov(Y,Y)(I-P_X)'=\sigma^2(I-P_X)$.
The matrix $\sigma^2(I-P_X)$ is a projection matrix, its eigenvalues are 0 or +1, it's positive semidefinite. So it should have non-negative values on the diagonal. So the scatter-plot of residuals against original dependent variable should show positive correlation.
As far as i know Gretl produces by default the graph of residuals against original dependent variable (not the predicted one!).
A: Is it possible you are confusing fitted/predicted values with the actual values?
As @gung and @biostat have said, you hope there is no relationship between fitted values and residuals.  On the other hand, finding a linear relationship between the actual values of the dependent/outcome variable and the residuals is to be expected and is not particularly informative.
Added to clarify the previous sentence:
Not just any linear relationship between residuals and actual values of the out come is to be expected... For low measured values of Y, the predicted values of Y from a useful model will tend to be higher than the actual measured values, and vice versa.
A: The answers offered are giving me some ideas about what's going on here. I do believe there may have been some mistakes made by accident.  See if the following story makes sense: To start, I think there is probably a strong relationship between X & Y in the data (here's some code and a plot):
set.seed(5)
wage <- rlnorm(1000, meanlog=2.3, sdlog=.5)
something_else <- .7*wage + rnorm(1000, mean=0, sd=1)
plot(wage, something_else, pch=3, col="red", main="Plot X vs. Y")


But by mistake Y was predicted just from the mean.  Compounding this, the residuals from the mean only model are plotted against X, even though what was intended was to plot against the fitted values (code & plot):
meanModel <- lm(something_else~1)
windows()
plot(wage, meanModel$residuals, pch=3, col="red", 
    main="Plot of residuals from Mean only Model against X")
abline(h=0, lty="dotted")


We can fix this by fitting the appropriate model and plotting the residuals from that (code & plot):
appropriateModel <- lm(something_else~wage)
windows()
plot(appropriateModel$fitted.values, appropriateModel$residuals, pch=3, col="red",
main="Plot of residuals from the appropriate\nmodel against fitted values")
lines(lowess(appropriateModel$residuals~appropriateModel$fitted.values))


This seems like just the kinds of goof-ups I made when I was starting.
A: Suppose that you have the regression $y_i = \beta_0 + \beta_1 x_i + \epsilon_i$, where $\beta_1 \approx 0$. Then, $y_i - \beta_0 \approx \epsilon_i$. The higher the $y$ value, the bigger the residual. On the contrary, a plot of the residuals against $x$ should show no systematic relationship. Also, the predicted value $\hat{y}_i$ should be approximately $\hat{\beta}_0$---the same for every observation. If all the predicted values are roughly the same, they should be uncorrelated with the errors.
What the plot is telling me is that $x$ and $y$ are essentially unrelated (of course, there are better ways to show this). Let us know if your coefficient $\hat{\beta}_1$ is not close to 0.
As better diagnostics, use a plot of the residuals against the predicted wage or against the $x$ value. You should not observe a distinguishable pattern in these plots.
If you want a little R demonstration, here you go:
y      <- rnorm(100, 0, 5)
x      <- rnorm(100, 0, 2)
res    <- lm(y ~ x)$residuals
fitted <- lm(y ~ x)$fitted.values
plot(y, res)
plot(x, res)
plot(fitted, res)

A: This graph indicates that the model you fitted is not good.  As @gung said in the first comments on the main question that there should be no relationship between predicated response and residual. 
" an analyst should expect a regression model to err in predicting a response in a random fashion; the model should predict values higher than actual and lower than actual with equal probability. See this"
I would recommend first plot response vs independent variable to see the relationship between them. It might be reasonable to add polynomial terms in the model.
A: Isn't this what happens if there is no relationship between the X & Y variable?  From looking at this graph, it appears you are essentially predicting Y with it's mean.
A: I think OP plotted residuals vs. the original response variable (not the fitted response variable from the model). I see plots like this all the time, with nearly the same exact pattern. Make sure you plot residuals vs. fitted values, as I'm not sure what meaningful inference you could gather from residuals vs. original Y. But I could certainly be wrong. 
