Some Useful Techniques
There are serveral techniqies involving backgrounds that will assist you in achieving better images. Some of these are described below.
Try to concentrate your photographic attempts on those subjects that are settled on isolated perches. This is an obvious technique that will naturally avoid a confused background.
Avoid areas of heavy shadow in the background, this is a problem with subjects perched on trees or scrubland / bushes. The shadow will only add dark patches to your image that will distract the eye from the quality of the subject matter.
Avoid areas of bright colour in the background caused by flowers or light foliage. Although out of focus it will create a visual distraction that will spoil the overall effect of the image.
One of the greatest and most annoying spoilers are water / dew droplets in the image. When these are illuminated by the sun they will appear as bright sequins and can render an otherwise good image as readily discardable. There is very little you can do to avoid these other than to wait for the moisture to dry out.
A Tip to Consider
This might seem obvious but always use your depth of field preview button. Your view through the viewfinder may show a clear background; however a quick glance with the preview button will show up any dark areas that may spoil your images.
It might not be possible to achieve all of the above and some compromise may be necessary. Sometimes changing the angle of the shot will solve a composition issue at other times the only option is to refuse the opportunity and wait for better one to come along.